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SIGMA, 2017, Volume 13, 059, 22 pages (Mi sigma1259)  

This article is cited in 5 scientific papers (total in 5 papers)

Remarks on Contact and Jacobi Geometry

Andrew James Brucea, Katarzyna Grabowskab, Janusz Grabowskic

a Mathematics Research Unit, University of Luxembourg, Luxembourg
b Faculty of Physics, University of Warsaw, Poland
c Institute of Mathematics, Polish Academy of Sciences, Poland

Abstract: We present an approach to Jacobi and contact geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key concepts are Kirillov manifolds and linear Kirillov structures, i.e., homogeneous Poisson manifolds and, respectively, homogeneous linear Poisson manifolds. The difference with the existing literature is that the homogeneity of the Poisson structure is related to a principal $GL(1,{\mathbb R})$-bundle structure on the manifold and not just to a vector field. This allows for working with Jacobi bundle structures on nontrivial line bundles and drastically simplifies the picture of Jacobi and contact geometry. Our results easily reduce to various basic theorems of Jacobi and contact geometry when the principal bundle structure is trivial, while giving new insights into the theory.

Keywords: symplectic structures; contact structures; Poisson structures; Jacobi structures; principal bundles; Lie groupoids; symplectic groupoids.

Funding Agency Grant Number
National Science Centre (Narodowe Centrum Nauki) DEC-2012/06/A/ST1/00256
The research of K. Grabowska and J. Grabowski was funded by the Polish National Science Centre grant under the contract number DEC-2012/06/A/ST1/00256.


DOI: https://doi.org/10.3842/SIGMA.2017.059

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Full text: http://www.emis.de/.../059
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Bibliographic databases:

MSC: 53D05; 53D10; 53D17; 58E40; 58H05
Received: January 16, 2017; in final form July 17, 2017; Published online July 26, 2017
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Citation: Andrew James Bruce, Katarzyna Grabowska, Janusz Grabowski, “Remarks on Contact and Jacobi Geometry”, SIGMA, 13 (2017), 059, 22 pp.

Citation in format AMSBIB
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\by Andrew~James~Bruce, Katarzyna~Grabowska, Janusz~Grabowski
\paper Remarks on Contact and Jacobi Geometry
\jour SIGMA
\yr 2017
\vol 13
\papernumber 059
\totalpages 22
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\crossref{https://doi.org/10.3842/SIGMA.2017.059}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85026847762}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. Sergyeyev, “New integrable (3+1)-dimensional systems and contact geometry”, Lett. Math. Phys., 108:2 (2018), 359–376  crossref  mathscinet  zmath  isi
    2. A. K. Prikarpatskii, “On the Linearization Covering Technique and its Application to Integrable Nonlinear Differential Systems”, SIGMA, 14 (2018), 023, 15 pp.  mathnet  crossref
    3. K. Grabowska, J. Grabowski, “$n$-Tuple principal bundles”, Int. J. Geom. Methods Mod. Phys., 15:12 (2018), 1850211  crossref  mathscinet  zmath  isi  scopus
    4. Bravetti A., “Contact Geometry and Thermodynamics”, Int. J. Geom. Methods Mod. Phys., 16:1, SI (2019), 1940003  crossref  mathscinet  isi  scopus
    5. Das A., “Gauge Transformations of Jacobi Structures and Contact Groupoids”, Math. Phys. Anal. Geom., 22:2 (2019), 11  crossref  mathscinet  zmath  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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